At this stage I am trying to avoid to make a call to that web design company untill its absolutely required.
Can someone tell me how we can know the email address of domain admin contract so we try to get the email and click on verification link when we initiate request at godaddy/namecheap.
A whois lookup on Nominet's website (nominet.org.uk) will show who the current registrar is (shown as registrants agent).
Phone the current registrar and ask them what you need to do change registrars (some require faxed confirmation, emails etc). Once they've received your confirmation, they will change the IPS key/tag (=registrar key used by Nominet) to your new registrar, and your new registrar will have control of the domain.
If you get stuck, you can always pay Nominet £10+Vat to change registrars for you.